Real-rooted atlas

A038320 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(17 + 40\,x + 23\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 782\,x + 529\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02897825291600
1491319941269791216764000
2835214519969172868273562798412560000
3141985796049152598977035162630237864856436343102400000
424137569195940266662739135119552942012131107356565069861480358894096000000
5410338673388614860915773191413355670002454812005915539062165667176162707913404825447163840000000
6697575744175502315832357525672028967422406664163608201127017708181769041197906413788463056260792783109852816553600000000
7118587876497144398179028778144896886122466927136993250064109544862677337952665786109322710318835423803950588119815807479931801152661463262144000000000

All 15 rows

Technical details
Node id
n-b4265c5ad843d43f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b6c8acc80ab25843

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            289,             782,              529;
  1:           4913,           19941,            26979,             12167;
  2:          83521,          451996,           917286,            827356,            279841;
  3:        1419857,         9604915,         25989770,          35162630,          23786485,           6436343;
  4:       24137569,       195940266,        662739135,        1195529420,        1213110735,         656506986,         148035889;
  5:      410338673,      3886148609,      15773191413,       35567000245,       48120059155,       39062165667,       17616270791,        3404825447;
  6:     6975757441,     75502315832,     357525672028,      967422406664,     1636082011270,     1770818176904,     1197906413788,      463056260792,       78310985281;
  7:   118587876497,   1443981790287,    7814489688612,    24669271369932,    50064109544862,    67733795266578,    61093227103188,    35423803950588,    11981580747993,     1801152661463;
  8:  2015993900449,  27275211594310,  166057905883005,   599110876126920,  1418483103771090,  2302949039063652,  2596462151885490,  2007348890533320,  1018434363579405,   306195952448710,   41426511213649;
  9: 34271896307633, 510046456813597, 3450314266680215, 14004216729466755, 37893762915027690, 71775245050817154, 97107684480517326, 93843560632432710, 63482408663116245, 28629321553954385, 7746757596952363, 952809757913927;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (17+40*x+23*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 782*x + 529*x^2.
%t Mathematica
P[0, x_] := 289 + 782*x + 529*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((17 + 40*x + 23*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 289 + 782*x + 529*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((17 + 40*x + 23*x^2)*v[i - 1])/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.